共查询到20条相似文献,搜索用时 31 毫秒
1.
David Ishii Smyth 《Inventiones Mathematicae》2013,192(2):459-503
The moduli space of smooth curves admits a beautiful compactification $\mathcal{M}_{g,n} \subset \overline{\mathcal{M}}_{g,n}$ by the moduli space of stable curves. In this paper, we undertake a systematic classification of alternate modular compactifications of $\mathcal{M}_{g,n}$ . Let $\mathcal{U}_{g,n}$ be the (non-separated) moduli stack of all n-pointed reduced, connected, complete, one-dimensional schemes of arithmetic genus g. When g=0, $\mathcal{U}_{0,n}$ is irreducible and we classify all open proper substacks of $\mathcal{U}_{0,n}$ . When g≥1, $\mathcal{U}_{g,n}$ may not be irreducible, but there is a unique irreducible component $\mathcal{V}_{g,n} \subset\mathcal{U}_{g,n}$ containing $\mathcal{M}_{g,n}$ . We classify open proper substacks of $\mathcal {V}_{g,n}$ satisfying a certain stability condition. 相似文献
2.
Sylwia Cichacz 《Graphs and Combinatorics》2014,30(3):565-571
A group distance magic labeling or a ${\mathcal{G}}$ -distance magic labeling of a graph G = (V, E) with ${|V | = n}$ is a bijection f from V to an Abelian group ${\mathcal{G}}$ of order n such that the weight ${w(x) = \sum_{y\in N_G(x)}f(y)}$ of every vertex ${x \in V}$ is equal to the same element ${\mu \in \mathcal{G}}$ , called the magic constant. In this paper we will show that if G is a graph of order n = 2 p (2k + 1) for some natural numbers p, k such that ${\deg(v)\equiv c \mod {2^{p+1}}}$ for some constant c for any ${v \in V(G)}$ , then there exists a ${\mathcal{G}}$ -distance magic labeling for any Abelian group ${\mathcal{G}}$ of order 4n for the composition G[C 4]. Moreover we prove that if ${\mathcal{G}}$ is an arbitrary Abelian group of order 4n such that ${\mathcal{G} \cong \mathbb{Z}_2 \times\mathbb{Z}_2 \times \mathcal{A}}$ for some Abelian group ${\mathcal{A}}$ of order n, then there exists a ${\mathcal{G}}$ -distance magic labeling for any graph G[C 4], where G is a graph of order n and n is an arbitrary natural number. 相似文献
3.
V. E. Maiorov 《Constructive Approximation》2013,37(2):283-293
Let ${\mathcal {P}_{n}^{d}}$ denote the space of polynomials on ? d of total degree n. In this work, we introduce the space of polynomials ${\mathcal {Q}_{2 n}^{d}}$ such that ${\mathcal {P}_{n}^{d}}\subset {\mathcal {Q}_{2 n}^{d}}\subset\mathcal{P}_{2n}^{d}$ and which satisfy the following statement: Let h be any fixed univariate even polynomial of degree n and $\mathcal{A}$ be a finite set in ? d . Then every polynomial P from the space ${\mathcal {Q}_{2 n}^{d}}$ may be represented by a linear combination of radial basis functions of the form h(∥x+a∥), $a\in \mathcal{A}$ , if and only if the set $\mathcal{A}$ is a uniqueness set for the space ${\mathcal {Q}_{2 n}^{d}}$ . 相似文献
4.
Hongping Ma 《Journal of Applied Mathematics and Computing》2014,45(1-2):111-135
The energy of a graph is defined as the sum of the absolute values of all eigenvalues of the graph. A tree is said to be non-starlike if it has at least two vertices with degree more than 2. A caterpillar is a tree in which a removal of all pendent vertices makes a path. Let $\mathcal{T}_{n,d}$ , $\mathbb{T}_{n,p}$ be the set of all trees of order n with diameter d, p pendent vertices respectively. In this paper, we investigate the relations on the ordering of trees and non-starlike trees by minimal energies between $\mathcal{T}_{n,d}$ and $\mathbb{T}_{n,n-d+1}$ . We first show that the first two trees (non-starlike trees, resp.) with minimal energies in $\mathcal{T}_{n,d}$ and $\mathbb{T}_{n,n-d+1}$ are the same for 3≤d≤n?2 (3≤d≤n?3, resp.). Then we obtain that the trees with third-minimal energy in $\mathcal{T}_{n,d}$ and $\mathbb{T}_{n,n-d+1}$ are the same when n≥11, 3≤d≤n?2 and d≠8; and the tree with third-minimal energy in $\mathcal{T}_{n,8}$ is the caterpillar with third-minimal energy in $\mathbb{T}_{n,n-7}$ for n≥11. 相似文献
5.
We study the structure of a metric n-Lie algebra G over the complex field C. Let G = SR be the Levi decomposition, where R is the radical of G and S is a strong semisimple subalgebra of G. Denote by m(G) the number of all minimal ideals of an indecomposable metric n-Lie algebra and R ⊥ the orthogonal complement of R. We obtain the following results. As S-modules, R ⊥ is isomorphic to the dual module of G/R. The dimension of the vector space spanned by all nondegenerate invariant symmetric bilinear forms on G is equal to that of the vector space of certain linear transformations on G; this dimension is greater than or equal to m(G) + 1. The centralizer of R in G is equal to the sum of all minimal ideals; it is the direct sum of R ⊥ and the center of G. Finally, G has no strong semisimple ideals if and only if R⊥■R. 相似文献
6.
Jun Cao Der-Chen Chang Dachun Yang Sibei Yang 《Integral Equations and Operator Theory》2013,76(2):225-283
Let ${\Phi}$ be a continuous, strictly increasing and concave function on (0, ∞) of critical lower type index ${p_\Phi^- \in(0,\,1]}$ . Let L be an injective operator of type ω having a bounded H ∞ functional calculus and satisfying the k-Davies–Gaffney estimates with ${k \in {\mathbb Z}_+}$ . In this paper, the authors first introduce an Orlicz–Hardy space ${H^{\Phi}_{L}(\mathbb{R}^n)}$ in terms of the non-tangential L-adapted square function and then establish its molecular characterization. As applications, the authors prove that the generalized Riesz transform ${D_{\gamma}L^{-\delta/(2k)}}$ is bounded from the Orlicz–Hardy space ${H^{\Phi}_{L}(\mathbb{R}^n)}$ to the Orlicz space ${L^{\widetilde{\Phi}}(\mathbb{R}^n)}$ when ${p_\Phi^- \in (0, \frac{n}{n+ \delta - \gamma}]}$ , ${0 < \gamma \le \delta < \infty}$ and ${\delta- \gamma < n (\frac{1}{p_-(L)}-\frac{1}{p_+(L)})}$ , or from ${H^{\Phi}_{L}(\mathbb{R}^n)}$ to the Orlicz–Hardy space ${H^{\widetilde \Phi}(\mathbb{R}^n)}$ when ${p_\Phi^-\in (\frac{n}{n + \delta+ \lfloor \gamma \rfloor- \gamma},\,\frac{n}{n+ \delta- \gamma}]}$ , ${1\le \gamma \le \delta < \infty}$ and ${\delta- \gamma < n (\frac{1}{p_-(L)}-\frac{1}{p_+(L)})}$ , or from ${H^{\Phi}_{L}(\mathbb{R}^n)}$ to the weak Orlicz–Hardy space ${WH^\Phi(\mathbb{R}^n)}$ when ${\gamma = \delta}$ and ${p_\Phi=n/(n + \lfloor \gamma \rfloor)}$ or ${p_\Phi^-=n/(n + \lfloor \gamma \rfloor)}$ with ${p_\Phi^-}$ attainable, where ${\widetilde{\Phi}}$ is an Orlicz function whose inverse function ${\widetilde{\Phi}^{-1}}$ is defined by ${\widetilde{\Phi}^{-1}(t):=\Phi^{-1}(t)t^{\frac{1}{n}(\gamma- \delta)}}$ for all ${t \in (0,\,\infty)}$ , ${p_\Phi}$ denotes the strictly critical lower type index of ${\Phi}$ , ${\lfloor \gamma \rfloor}$ the maximal integer not more than ${\gamma}$ and ${(p_-(L),\,p_+(L))}$ the range of exponents ${p \in[1,\, \infty]}$ for which the semigroup ${\{e^{-tL}\}_{t >0 }}$ is bounded on ${L^p(\mathbb{R}^n)}$ . 相似文献
7.
Toshio Sumi Mitsuhiro Miyazaki Toshio Sakata 《Annals of the Institute of Statistical Mathematics》2010,62(4):807-822
Tensor data are becoming important recently in various application fields. In this paper, we consider the maximal rank problem of 3-tensors and extend Atkinson and Stephens’ and Atkinson and Lloyd’s results over the real number field. We also prove the assertion of Atkinson and Stephens: ${{\rm max.rank}_{\mathbb{R}}(m,n,p) \leq m+\lfloor p/2\rfloor n}$ , ${{\rm max.rank}_{\mathbb{R}}(n,n,p) \leq (p+1)n/2}$ if p is even, ${{\rm max.rank}_{\mathbb{F}}(n,n,3)\leq 2n-1}$ if ${\mathbb{F}=\mathbb{C}}$ or n is odd, and ${{\rm max.rank}_{\mathbb{F}}(m,n,3)\leq m+n-1}$ if m < n where ${\mathbb{F}}$ stands for ${\mathbb{R}}$ or ${\mathbb{C}}$ . 相似文献
8.
Let (M,g) be an n-dimensional, compact Riemannian manifold and ${P_0(\hbar) = -\hbar{^2} \Delta_g + V(x)}$ be a semiclassical Schrödinger operator with ${\hbar \in (0,\hbar_0]}$ . Let ${E(\hbar) \in [E-o(1),E+o(1)]}$ and ${(\phi_{\hbar})_{\hbar \in (0,\hbar_0]}}$ be a family of L 2-normalized eigenfunctions of ${P_0(\hbar)}$ with ${P_0(\hbar) \phi_{\hbar} = E(\hbar) \phi_{\hbar}}$ . We consider magnetic deformations of ${P_0(\hbar)}$ of the form ${P_u(\hbar) = - \Delta_{\omega_u}(\hbar) + V(x)}$ , where ${\Delta_{\omega_u}(\hbar) = (\hbar d + i \omega_u(x))^*({\hbar}d + i \omega_u(x))}$ . Here, u is a k-dimensional parameter running over ${B^k(\epsilon)}$ (the ball of radius ${\epsilon}$ ), and the family of the magnetic potentials ${(w_u)_{u\in B^k(\epsilon)}}$ satisfies the admissibility condition given in Definition 1.1. This condition implies that k ≥ n and is generic under this assumption. Consider the corresponding family of deformations of ${(\phi_{\hbar})_{\hbar \in (0, \hbar_0]}}$ , given by ${(\phi^u_{\hbar})_{\hbar \in(0, \hbar_0]}}$ , where $$\phi_{\hbar}^{(u)}:= {\rm e}^{-it_0 P_u(\hbar)/\hbar}\phi_{\hbar}$$ for ${|t_0|\in (0,\epsilon)}$ ; the latter functions are themselves eigenfunctions of the ${\hbar}$ -elliptic operators ${Q_u(\hbar): ={\rm e}^{-it_0P_u(\hbar)/\hbar} P_0(\hbar) {\rm e}^{it_0 P_u(\hbar)/\hbar}}$ with eigenvalue ${E(\hbar)}$ and ${Q_0(\hbar) = P_{0}(\hbar)}$ . Our main result, Theorem1.2, states that for ${\epsilon >0 }$ small, there are constants ${C_j=C_j(M,V,\omega,\epsilon) > 0}$ with j = 1,2 such that $$C_{1}\leq \int\limits_{\mathcal{B}^k(\epsilon)} |\phi_{\hbar}^{(u)}(x)|^2 \, {\rm d}u \leq C_{2}$$ , uniformly for ${x \in M}$ and ${\hbar \in (0,h_0]}$ . We also give an application to eigenfunction restriction bounds in Theorem 1.3. 相似文献
9.
Let ${N \geq 3}$ and u be the solution of u t = Δ log u in ${\mathbb{R}^N \times (0, T)}$ with initial value u 0 satisfying ${B_{k_1}(x, 0) \leq u_{0} \leq B_{k_2}(x, 0)}$ for some constants k 1 > k 2 > 0 where ${B_k(x, t) = 2(N - 2)(T - t)_{+}^{N/(N - 2)}/(k + (T - t)_{+}^{2/(N - 2)}|x|^{2})}$ is the Barenblatt solution for the equation and ${u_0 - B_{k_0} \in L^{1}(\mathbb{R}^{N})}$ for some constant k 0 > 0 if ${N \geq 4}$ . We give a new different proof on the uniform convergence and ${L^1(\mathbb{R}^N)}$ convergence of the rescaled function ${\tilde{u}(x, s) = (T - t)^{-N/(N - 2)}u(x/(T - t)^{-1/(N - 2)}, t), s = -{\rm log}(T - t)}$ , on ${\mathbb{R}^N}$ to the rescaled Barenblatt solution ${\tilde{B}_{k_0}(x) = 2(N - 2)/(k_0 + |x|^{2})}$ for some k 0 > 0 as ${s \rightarrow \infty}$ . When ${N \geq 4, 0 \leq u_0(x) \leq B_{k_0}(x, 0)}$ in ${\mathbb{R}^N}$ , and ${|u_0(x) - B_{k_0}(x, 0)| \leq f \in L^{1}(\mathbb{R}^{N})}$ for some constant k 0 > 0 and some radially symmetric function f, we also prove uniform convergence and convergence in some weighted L 1 space in ${\mathbb{R}^N}$ of the rescaled solution ${\tilde{u}(x, s)}$ to ${\tilde{B}_{k_0}(x)}$ as ${s \rightarrow \infty}$ . 相似文献
10.
Peter J. C. Dickinson Mirjam Dür Luuk Gijben Roland Hildebrand 《Optimization Letters》2013,7(8):1669-1679
We investigate the relation between the cone ${\mathcal{C}^{n}}$ of n × n copositive matrices and the approximating cone ${\mathcal{K}_{n}^{1}}$ introduced by Parrilo. While these cones are known to be equal for n ≤ 4, we show that for n ≥ 5 they are not equal. This result is based on the fact that ${\mathcal{K}_{n}^{1}}$ is not invariant under diagonal scaling. We show that for any copositive matrix which is not the sum of a nonnegative and a positive semidefinite matrix we can find a scaling which is not in ${\mathcal{K}_{n}^{1}}$ . In fact, we show that if all scaled versions of a matrix are contained in ${\mathcal{K}_{n}^{r}}$ for some fixed r, then the matrix must be in ${\mathcal{K}_{n}^{0}}$ . For the 5 × 5 case, we show the more surprising result that we can scale any copositive matrix X into ${\mathcal{K}_{5}^{1}}$ and in fact that any scaling D such that ${(DXD)_{ii} \in \{0,1\}}$ for all i yields ${DXD \in \mathcal{K}_{5}^{1}}$ . From this we are able to use the cone ${\mathcal{K}_{5}^{1}}$ to check if any order 5 matrix is copositive. Another consequence of this is a complete characterisation of ${\mathcal{C}^{5}}$ in terms of ${\mathcal{K}_{5}^{1}}$ . We end the paper by formulating several conjectures. 相似文献
11.
Let ${\mathcal{M}}$ be a Brakke flow of n-dimensional surfaces in ${\mathbb{R}^N}$ . The singular set ${\mathcal{S} \subset \mathcal{M}}$ has a stratification ${\mathcal{S}^0 \subset \mathcal{S}^1 \subset \cdots \mathcal{S}}$ , where ${X \in \mathcal{S}^j}$ if no tangent flow at X has more than j symmetries. Here, we define quantitative singular strata ${\mathcal{S}^j_{\eta, r}}$ satisfying ${\cup_{\eta>0} \cap_{0<r} \mathcal{S}^j_{\eta, r} = \mathcal{S}^j}$ . Sharpening the known parabolic Hausdorff dimension bound ${{\rm dim} \mathcal{S}^j \leq j}$ , we prove the effective Minkowski estimates that the volume of r-tubular neighborhoods of ${\mathcal{S}^j_{\eta, r}}$ satisfies ${{\rm Vol} (T_r(\mathcal{S}^j_{\eta, r}) \cap B_1) \leq Cr^{N + 2 - j-\varepsilon}}$ . Our primary application of this is to higher regularity of Brakke flows starting at k-convex smooth compact embedded hypersurfaces. To this end, we prove that for the flow of k-convex hypersurfaces, any backwards selfsimilar limit flow with at least k symmetries is in fact a static multiplicity one plane. Then, denoting by ${\mathcal{B}_r \subset \mathcal{M}}$ the set of points with regularity scale less than r, we prove that ${{\rm Vol}(T_r(\mathcal{B}_r)) \leq C r^{n+4-k-\varepsilon}}$ . This gives L p -estimates for the second fundamental form for any p < n + 1 ? k. In fact, the estimates are much stronger and give L p -estimates for the reciprocal of the regularity scale. These estimates are sharp. The key technique that we develop and apply is a parabolic version of the quantitative stratification method introduced in Cheeger and Naber (Invent. Math., (2)191 2013), 321–339) and Cheeger and Naber (Comm. Pure. Appl. Math, arXiv:1107.3097v1, 2013). 相似文献
12.
We provide a sufficient condition for the nontriviality of the Lipschitz homotopy group of the Heisenberg group, ${\pi_m^{\rm Lip}(\mathbb{H}_n)}$ , in terms of properties of the classical homotopy group of the sphere, ${\pi_m(\mathbb{S}^n)}$ . As an application we provide a new simplified proof of the fact that ${\pi_n^{\rm Lip}(\mathbb{H}_n)\neq \{0\}, n=1,2,\ldots}$ , and we prove a new result that ${\pi_{4n-1}^{\rm Lip}(\mathbb{H}_{2n})\neq \{0\}}$ for n = 1,2,… The last result is based on a new generalization of the Hopf invariant. We also prove that Lipschitz mappings are not dense in the Sobolev space ${W^{1,p}(\mathcal{M},\mathbb{H}_{2n})}$ when ${\dim \mathcal{M} \geq 4n}$ and 4n?1 ≤ p < 4n. 相似文献
13.
Imre Kocsis 《Results in Mathematics》2012,62(1-2):67-71
In this note we prove the following: Let n?≥ 2 be a fixed integer. A system of additive functions ${A_{1},A_{2},\ldots,A_{n}:\mathbb{R} \to\mathbb{R}}$ is linearly dependent (as elements of the ${\mathbb{R}}$ vector space ${\mathbb{R}^{\mathbb{R}}}$ ), if and only if, there exists an indefinite quadratic form ${Q:\mathbb{R}^{n}\to\mathbb{R} }$ such that ${Q(A_{1}(x),A_{2}(x),\ldots,A_{n}(x))\geq 0}$ or ${Q(A_{1}(x),A_{2}(x),\ldots,A_{n}(x))\leq 0}$ holds for all ${x\in\mathbb{R}}$ . 相似文献
14.
Bart De Bruyn 《Designs, Codes and Cryptography》2012,64(1-2):81-91
Let ${{\mathbb H}_n, n \geq 1}$ , be the near 2n-gon defined on the 1-factors of the complete graph on 2n?+?2 vertices, and let e denote the absolutely universal embedding of ${{\mathbb H}_n}$ into PG(W), where W is a ${\frac{1}{n+2} \left(\begin{array}{c}2n+2 \\ n+1\end{array}\right)}$ -dimensional vector space over the field ${{\mathbb F}_2}$ with two elements. For every point z of ${{\mathbb H}_n}$ and every ${i \in {\mathbb N}}$ , let Δ i (z) denote the set of points of ${{\mathbb H}_n}$ at distance i from z. We show that for every pair {x, y} of mutually opposite points of ${{\mathbb H}_n, W}$ can be written as a direct sum ${W_0 \oplus W_1 \oplus \cdots \oplus W_n}$ such that the following four properties hold for every ${i \in \{0,\ldots,n \}}$ : (1) ${\langle e(\Delta_i(x) \cap \Delta_{n-i}(y)) \rangle = {\rm PG}(W_i)}$ ; (2) ${\left\langle e \left( \bigcup_{j \leq i} \Delta_j(x) \right) \right\rangle = {\rm PG}(W_0 \oplus W_1 \oplus \cdots \oplus W_i)}$ ; (3) ${\left\langle e \left( \bigcup_{j \leq i} \Delta_j(y) \right) \right\rangle = {\rm PG}(W_{n-i}\oplus W_{n-i+1} \oplus \cdots \oplus W_n)}$ ; (4) ${\dim(W_i) = |\Delta_i(x) \cap \Delta_{n-i}(y)| = \left(\begin{array}{c}n \\ i\end{array}\right)^2 - \left(\begin{array}{c}n \\ i-1\end{array}\right) \cdot \left(\begin{array}{c}n \\ i+1\end{array}\right)}$ . 相似文献
15.
V. N. Potapov 《Mathematical Notes》2013,93(3-4):479-486
Let Σ be a finite set of cardinality k > 0, let $\mathbb{A}$ be a finite or infinite set of indices, and let $\mathcal{F} \subseteq \Sigma ^\mathbb{A}$ be a subset consisting of finitely supported families. A function $f:\Sigma ^\mathbb{A} \to \Sigma$ is referred to as an $\mathbb{A}$ -quasigroup (if $\left| \mathbb{A} \right| = n$ , then an n-ary quasigroup) of order k if $f\left( {\bar y} \right) \ne f\left( {\bar z} \right)$ for any ordered families $\bar y$ and $\bar z$ that differ at exactly one position. It is proved that an $\mathbb{A}$ -quasigroup f of order 4 is reducible (representable as a superposition) or semilinear on every coset of $\mathcal{F}$ . It is shown that the quasigroups defined on Σ?, where ? are positive integers, generate Lebesgue nonmeasurable subsets of the interval [0, 1]. 相似文献
16.
Trieu Le 《Integral Equations and Operator Theory》2010,67(3):425-438
Let ${\vartheta}$ be a measure on the polydisc ${\mathbb{D}^n}$ which is the product of n regular Borel probability measures so that ${\vartheta([r,1)^n\times\mathbb{T}^n) >0 }$ for all 0 < r < 1. The Bergman space ${A^2_{\vartheta}}$ consists of all holomorphic functions that are square integrable with respect to ${\vartheta}$ . In one dimension, it is well known that if f is continuous on the closed disc ${\overline{\mathbb{D}}}$ , then the Hankel operator H f is compact on ${A^2_\vartheta}$ . In this paper we show that for n ≥ 2 and f a continuous function on ${{\overline{\mathbb{D}}}^n}$ , H f is compact on ${A^2_\vartheta}$ if and only if there is a decomposition f = h + g, where h belongs to ${A^2_\vartheta}$ and ${\lim_{z\to\partial\mathbb{D}^n}g(z)=0}$ . 相似文献
17.
Richard Nickl 《Probability Theory and Related Fields》2007,138(3-4):411-449
Let ${\mathcal{P}}$ be a nonparametric probability model consisting of smooth probability densities and let ${\hat{p}_{n}}$ be the corresponding maximum likelihood estimator based on n independent observations each distributed according to the law ${\mathbb{P}}$ . With $\hat{\mathbb{P}}_{n}$ denoting the measure induced by the density ${\hat{p}_{n}}$ , define the stochastic process ${\hat{\nu}}_{n}: f\longmapsto \sqrt{n} \int fd({\hat{\mathbb{P}}}_{n} -\mathbb{P})$ where f ranges over some function class ${\mathcal{F}}$ . We give a general condition for Donsker classes ${\mathcal{F}}$ implying that the stochastic process $\hat{\nu}_{n}$ is asymptotically equivalent to the empirical process in the space ${\ell ^{\infty }(\mathcal{F})}$ of bounded functions on ${ \mathcal{F}}$ . This implies in particular that $\hat{\nu}_{n}$ converges in law in ${\ell ^{\infty }(\mathcal{F})}$ to a mean zero Gaussian process. We verify the general condition for a large family of Donsker classes ${\mathcal{ F}}$ . We give a number of applications: convergence of the probability measure ${\hat{\mathbb{P}}_{n}}$ to ${\mathbb{P}}$ at rate ${\sqrt{n}}$ in certain metrics metrizing the topology of weak(-star) convergence; a unified treatment of convergence rates of the MLE in a continuous scale of Sobolev-norms; ${\sqrt{n}}$ -efficient estimation of nonlinear functionals defined on ${\mathcal{P}}$ ; limit theorems at rate ${\sqrt{n}}$ for the maximum likelihood estimator of the convolution product ${\mathbb{P\ast P}}$ . 相似文献
18.
Andrew Lorent 《Calculus of Variations and Partial Differential Equations》2013,48(3-4):625-665
We consider the following question: Given a connected open domain ${\Omega \subset \mathbb{R}^n}$ , suppose ${u, v : \Omega \rightarrow \mathbb{R}^n}$ with det ${(\nabla u) > 0}$ , det ${(\nabla v) > 0}$ a.e. are such that ${\nabla u^T(x)\nabla u(x) = \nabla v(x)^T \nabla v(x)}$ a.e. , does this imply a global relation of the form ${\nabla v(x) = R\nabla u(x)}$ a.e. in Ω where ${R \in SO(n)}$ ? If u, v are C 1 it is an exercise to see this true, if ${u, v\in W^{1,1}}$ we show this is false. In Theorem 1 we prove this question has a positive answer if ${v \in W^{1,1}}$ and ${u \in W^{1,n}}$ is a mapping of L p integrable dilatation for p > n ? 1. These conditions are sharp in two dimensions and this result represents a generalization of the corollary to Liouville’s theorem that states that the differential inclusion ${\nabla u \in SO(n)}$ can only be satisfied by an affine mapping. Liouville’s corollary for rotations has been generalized by Reshetnyak who proved convergence of gradients to a fixed rotation for any weakly converging sequence ${v_k \in W^{1,1}}$ for which $$\int \limits_{\Omega} {\rm dist}(\nabla v_k, SO(n))dz \rightarrow 0 \, {\rm as} \, k \rightarrow \infty.$$ Let S(·) denote the (multiplicative) symmetric part of a matrix. In Theorem 3 we prove an analogous result to Theorem 1 for any pair of weakly converging sequences ${v_k \in W^{1,p}}$ and ${u_k \in W^{1,\frac{p(n-1)}{p-1}}}$ (where ${p \in [1, n]}$ and the sequence (u k ) has its dilatation pointwise bounded above by an L r integrable function, r > n ? 1) that satisfy ${\int_{\Omega} |S(\nabla u_k) - S(\nabla v_k)|^p dz \rightarrow 0}$ as k → ∞ and for which the sign of the det ${(\nabla v_k)}$ tends to 1 in L 1. This result contains Reshetnyak’s theorem as the special case (u k ) ≡ Id, p = 1. 相似文献
19.
Let $\mathfrak{F}_{\lambda}^{n}$ be the $\mathop {\mathfrak {osp}}\nolimits \,(n|2)$ -module of weighted densities on ?1|n of weight ??. We compute the cohomology spaces $\mathrm{H}^{k}_{\mathrm{diff}}\left(\mathop {\mathfrak {osp}}\nolimits \,(n|2),\mathfrak{F}_{\lambda}^{n}\right)$ , where k=1 and n=0,1,2 or k=2 and n=0,1. We explicitly give cocycles spanning these cohomology spaces. 相似文献
20.
Daomin Cao Shusen Yan 《Calculus of Variations and Partial Differential Equations》2010,38(3-4):471-501
In this paper, we will prove the existence of infinitely many solutions for the following elliptic problem with critical Sobolev growth and a Hardy potential: $$-\Delta u-\frac{\mu}{|x|^2}u = |u|^{2^{\ast}-2}u+a u\quad {\rm in}\;\Omega,\quad u=0 \quad {\rm on}\; \partial\Omega,\qquad (*)$$ under the assumptions that N ≥ 7, ${\mu\in \left[0,\frac{(N-2)^2}4-4\right)}$ and a > 0, where ${2^{\ast}=\frac{2N}{N-2}}$ , and Ω is an open bounded domain in ${\mathbb{R}^N}$ which contains the origin. To achieve this goal, we consider the following perturbed problem of (*), which is of subcritical growth, $$-\Delta u-\frac{\mu}{|x|^2}u = |u|^{2^{\ast}-2-\varepsilon_n}u+au \quad {\rm in}\,\Omega, \quad u=0 \quad {\rm on}\;\partial\Omega,\qquad(\ast\ast)_n$$ where ${\varepsilon_{n} > 0}$ is small and ${\varepsilon_n \to 0}$ as n → + ∞. By the critical point theory for the even functionals, for each fixed ${\varepsilon_{n} > 0}$ small, (**) n has a sequence of solutions ${u_{k,\varepsilon_{n}} \in H^{1}_{0}(\Omega)}$ . We obtain the existence of infinitely many solutions for (*) by showing that as n → ∞, ${u_{k,\varepsilon_{n}}}$ converges strongly in ${H^{1}_{0}(\Omega)}$ to u k , which must be a solution of (*). Such a convergence is obtained by applying a local Pohozaev identity to exclude the possibility of the concentration of ${\{u_{k,\varepsilon_n}\}}$ . 相似文献